PROFESSOR CAYLEY’S TENTH MEMOIR ON QUANTICS. 
639 
order of either of the component co variants, then also the derivative is =0 : in 
particular neither of the covariants must be an invariant. The degree of the 
derivative is evidently equal to the sum of the degrees of the component covariants ; 
the order is equal to the sum of the orders less twice the index of derivation. 
383. It was by means of the theory of derivatives that Gordan proved (for a 
binary quantic of any order) that the number of covariants was finite, and in the 
particular case of the quintic, established the system of the 23 co variants. Starting 
from the quantic itself a, then the system of derivatives act 2, aoA, &c., must include 
among itself all the covariants of the second degree, and if the entire system of these 
is, suppose, b, c, then the derivatives ah 1, ctb2, ab3, &c., act, ac2, &c., must include 
among them all the covariants of the third degree, and so on for the higher degrees ; 
and in this way limiting by general reasoning the number of the independent 
covariants of each degree obtained by the successive steps, the foregoing conclusion is 
arrived at. But returning to the quintic, and supposing the system of the 23 
covariants established, then knowing the deg-orcler of a derivative we know that it 
must be a linear function of the segregates of that cleg-order ; and we thus confirm, 
a posteriori, the results of the derivation theory. I annex the following Table No. 99, 
which shows all the derivatives which present themselves, and for each of them the 
covariants as well congregate as segregate of the same deg-order : the congregates 
are distinguished each by two prefixed dots, . . bf, &c. No further explanation of 
the arrangement is, I think, required. We see from the table in what manner 
the different covariants present themselves in connexion with the derivation-theory. 
Thus starting with the quintic itself a, we have the two derivatives ctai, aa2, 
winch are in fact the covariants of the second degree (cleg-orders 2.2 and 2.6 respec- 
tively) b and c. For the third degree we have the derivatives ab2, ab 1, etc 5, ac 4, 
ctc3, ctc2, acl : the cleg-order of ctc5 is 3.1, and there being no covariants of this 
deg-order, ac5 must, it is clear, vanish identically : ab2 and ac4 are each of them 
of the deg-order 3.3, but for this deg-order we have only the covariant cl, and hence 
ab2 and ac4 must be each of them a numerical multiple of d ; similarly, deg-orcler 
3.5, abl and «c3 must be each of them a numerical multiple of e ; deg-orcler 3.7, ctc2 
must be a numerical multiple of ab ; and deg-order 3.9, ctcl must be a numerical 
multiple of f : the 7 derivatives, which primd facie might give, each of them, a 
covariant of the third degree, thus give in fact only the 3 covariants d, e, f; and in 
order to show according to the theory of derivations that this is so, it is necessary 
to prove — 1°, that «co = 0 ; 2°, that ac4 and ab2 differ only by a numerical factor; 
3°, that abl and «c3 differ only by a numerical factor ; 4°, that ac2 is a numerical 
multiple of ab : which being so we have the 3 new covariants. The table shows that 
for degrees 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22. 23, 24 
No. of derivatives = 2, 7, 19, 29, 41, 46, 52, 46, 44, 35, 26, 19, 17, 12, 13, 6, 6, 3, 3, 1, 1, 0, 1 
so that the whole number of derivatives is 429, giving the 22 co variants b, c . . . w. 
MDCCCLXXVHI. 4 N 
